Simulation of elastic wave propagation in geological media: Intercomparison of three numerical methods

V. A. Biryukov   ; V. A. Miryaha   ; I. B. Petrov   ; N. I. Khokhlov

Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 56 (2016) no. 6, pp. 1104-1114

Résumé

For wave propagation in heterogeneous media, we compare numerical results produced by grid-characteristic methods on structured rectangular and unstructured triangular meshes and by a discontinuous Galerkin method on unstructured triangular meshes as applied to the linear system of elasticity equations in the context of direct seismic exploration with an anticlinal trap model. It is shown that the resulting synthetic seismograms are in reasonable quantitative agreement. The grid-characteristic method on structured meshes requires more nodes for approximating curved boundaries, but it has a higher computation speed, which makes it preferable for the given class of problems.

Bibliographie

[1] Carcione J. M., Herman G. C., ten Kroode A. P. E., “Y2K review article: seismic modeling”, Review Literature And Arts Of The Americas, 67:4 (2002), 1304–1325

[2] Virieux J., Calandra H., Plessix R. É., “A review of the spectral, pseudo-spectral, finite-difference and finite-element modelling techniques for geophysical imaging”, Geophysical Prospecting, 59:5 (2011), 794–813 | DOI

[3] Virieux J., Etienne V. et al., “Modelling seismic wave propagation for geophysical imaging”, Seismic Waves — Research and Analysis, InTech (2012), 253–304

[4] Moczo P., Kristek J., Galis M., The finite-difference modelling of earthquake motions: waves and ruptures, Cambridge Univ. Press, 2014 | Zbl

[5] Novatskii V., Teoriya uprugosti, Mir, M., 1975

[6] Etienne V., Chaljub E. et al., “An hp-adaptive discontinuous Galerkin finite-element method for 3D elastic wave modelling”, Geophys. J. Int., 183:2 (2010), 941–962 | DOI

[7] Hermann V., Käser M., Castro C. E., “Non-conforming hybrid meshes for efficient 2-D wave propagation using the discontinuous Galerkin method”, Geophysical Journal International, 184:2 (2011), 746–758 | DOI

[8] Miryakha V. A., Sannikov A. V., Petrov I. B., “Chislennoe modelirovanie dinamicheskikh protsessov v tverdykh deformiruemykh telakh razryvnym metodom Galerkina”, Matem. modelirovanie, 27:3 (2015), 96–108 | Zbl

[9] Mercerat E. D., Glinsky N., “A nodal high-order discontinuous Galerkin method for elastic wave propagation in arbitrary heterogeneous media”, Geophysical J. Internat., 201:2 (2015), 1099–1116 | DOI

[10] Chelnokov F. B., “Yavnoe predstavlenie setochno-kharakteristicheskikh skhem dlya uravnenii uprugosti v dvumernom i trekhmernom prostranstvakh”, Matem. modelirovanie, 18:6 (2006), 96–108 | MR | Zbl

[11] Kvasov I. E., Pankratov S. A., Petrov I. B., “Numerical simulation of seismic responses in multilayer geologic media by the grid-characteristic method”, Mathematical Models and Computer Simulations, 3:2 (2011), 196–204 | DOI | MR

[12] Petrov I. B., Favorskaya A. V. et al., “Grid-characteristic method using high-order interpolation on tetrahedral hierarchical meshes with a multiple time step”, Mathematical Models and Computer Simulations, 5:5 (2013), 409–415 | DOI

[13] Golubev V. I., Petrov I. B., Khokhlov N. I., “Modelirovanie volnovykh protsessov vnutri planety s pomoschyu gibridnogo setochno-kharakteristicheskogo metoda”, Matem. modelirovanie, 27:2 (2015), 139–148 | Zbl

[14] Golubev V. I., Petrov I. B. et al., “Numerical computation of wave propagation in fractured media by applying the grid-characteristic method on hexahedral meshes”, Computat. Math. and Math. Phys., 55:3 (2015), 509–518 | DOI | MR | Zbl

[15] Petrov I. B., Favorskaya A. V. et al., “Monitoring the state of the moving train by use of high performance systems and modern computation methods”, Mathematical Models and Computer Simulations, 7:1 (2015), 51–61 | DOI

[16] Kvasov I. E., Petrov I. B., “High-performance computer simulation of wave processes in geological media in seismic exploration”, Computat. Math. and Math. Phys., 52:2 (2012), 302–313 | DOI | MR | Zbl

[17] Kvasov I. E., Levyant V. B., Petrov I. B., “Chislennoe modelirovanie pryamykh otklikov ot plastovykh zon s subvertikalnymi flyuidonasyschennymi mezotreschinami”, Tekhnologii seismorazvedki, 2013, no. 3, 19–35 | MR

[18] Golubev V. I., Petrov I. B., Khokhlov N. I., “Numerical simulation of seismic activity by the grid-characteristic method”, Computat. Math. and Math. Phys., 53:10 (2013), 1523–1533 | DOI | MR

[19] Petrov I. B., Khokhlov N. I., “Modeling 3D seismic problems using high-performance computing systems”, Mathematical Models and Computer Simulations, 6:4 (2014), 342–350 | DOI | MR

[20] Versteeg R., “The Marmousi experience: velocity model determination on a synthetic complex data set”, The Leading Edge, 13 (1994), 927–936 | DOI

[21] Martin G. S., Wiley R., Marfurt K. J., “Marmousi2: An Elastic Upgrade for Marmousi”, The Leading Edge, 25:2 (2006), 156–166 | DOI

[22] De Basabe J. D., Sen Mrinal K., “New developments in the finite-element method for seismic modeling”, The Leading Edge, 28:5 (2009), 562–567 | DOI

[23] Castro C. E., Käser M., Brietzke G. B., “Seismic waves in heterogeneous material: Subcell resolution of the discontinuous Galerkin method”, Geophysical J. Internat., 182:1 (2010), 250–264

[24] Wenk S., Pelties C. et al., “Regional wave propagation using the discontinuous Galerkin method”, Solid Earth, 4:1 (2013), 43–57 | DOI

[25] Robertsson J., Laws R. et al., “Modelling of scattering of seismic waves from a corrugated rough sea surface: a comparison of three methods”, Geophysical J. Internat., 167:1 (2006), 70–76 | DOI

[26] Moczo P., Kristek J. et al., “3-D finite-difference, finite-element, discontinuous-Galerkin and spectral-element schemes analysed for their accuracy with respect to P-wave to S-wave speed ratio”, Geophysical J. Internat., 187:3 (2011), 1645–1667 | DOI

[27] LeVeque R. J., Finite volume methods for hyperbolic problems, Cambridge University Press, Cambridge, New York, 2002 | MR | Zbl

[28] Käser M., Dumbser M., “An arbitrary high-order discontinuous Galerkin method for elastic waves on unstructured meshes. I. The two-dimensional isotropic case with external source terms”, Geophysical J. Internat., 166:2 (2006), 855–877 | DOI

[29] Miryakha V. A., Sannikov A. V., Petrov I. B., “Chislennoe modelirovanie dinamicheskikh protsessov v tverdykh deformiruemykh telakh razryvnym metodom Galerkina”, Matem. modelirovanie, 27:3 (2015), 96–108 | Zbl

[30] Wilcox L., Stadler G. et al., “A high-order discontinuous Galerkin method for wave propagation through coupled elastic-acoustic media”, Journal Comput. Phys., 229:24 (2010), 9373–9396 | DOI | MR | Zbl

[31] Agapov P. I., Chelnokov F. B., “Sravnitelnyi analiz raznostnykh skhem dlya chislennogo resheniya dvumernykh zadach mekhaniki deformiruemogo tverdogo tela”, Modelirovanie i obrabotka informatsii, Sb. st., 2003, 19–27

[32] Kholodov A. S., Kholodov Ya. A., “O kriteriyakh monotonnosti raznostnykh skhem dlya uravnenii giperbolicheskogo tipa”, Zh. vychisl. matem. i matem. fiz., 46:9 (2006), 1638–1667 | MR

[33] Kvasov I. E., Pankratov S. A., Petrov I. B., “Chislennoe modelirovanie seismicheskikh otklikov v mnogosloinykh geologicheskikh sredakh setochno-kharakteristicheskim metodom”, Matem. modelirovanie, 22:9 (2010), 13–22

[34] Golubev V. I., Petrov I. B., Khokhlov N. I., “Chislennoe modelirovanie seismicheskoi aktivnosti setochno-kharakteristicheskim metodom”, Zh. vychisl. matem. i matem. fiz., 53:10 (2013), 1709–1720 | DOI | MR | Zbl

[35] Kulikovskii A. G., Pogorelov N. V., Semenov A. Yu., Matematicheskie voprosy chislennogo resheniya giperbolicheskikh sistem uravnenii, Fizmatlit, M., 2001

[36] Komatitsch D., “The spectral-element method in seismology”, Geophysical Monograph Series, 157 (2005), 205–227

[37] De Basabe J. D., Sen Mrinal K., “Grid dispersion and stability criteria of some common finite-element methods for acoustic and elastic wave equations”, Geophysics, 72:6 (2007), T81–T95 | DOI

[38] Käser M., Dumbser M., “A highly accurate discontinuous Galerkin method for complex interfaces between solids and moving fluids”, Geophysics, 73:3 (2008), T23 | DOI

[39] Pelties C., Käser M., “Regular versus irregular meshing for complicated models and their effect on synthetic seismograms”, Geophys. J. Int., 183:2 (2010), 1031–1051 | DOI

[40] Bonoa G., Awruchb M. A., “Numerical study between structured and unstructured meshes for Euler and Navier–Stokes equations”, Mecanica Computational, 26 (2007), 3134–3146